Question

10£®ÒÑÖª£º$\frac{1}{1¡Á2}$=1-$\frac{1}{2}$£»$\frac{1}{2¡Á3}$=$\frac{1}{2}-\frac{1}{3}$£»$\frac{1}{3¡Á4}$=$\frac{1}{3}-\frac{1}{4}$£»$\frac{1}{4¡Á5}$=$\frac{1}{4}$-$\frac{1}{5}$£»¡­
£¨1£©Ìî¿Õ£º$\frac{1}{2}$+$\frac{1}{2¡Á3}$+$\frac{1}{3¡Á4}$+¡­+$\frac{1}{n£¨n+1£©}$=$\frac{n}{n+1}$£»
£¨2£©¸ù¾ÝÄã·¢ÏֵĹæÂɽⷽ³Ì£º
$\frac{1}{£¨x+2£©£¨x+3£©}$+$\frac{1}{£¨x+3£©£¨x+4£©}$+$\frac{1}{£¨x+4£©£¨x+5£©}$+¡­+$\frac{1}{£¨x+2013£©£¨x+2014£©}$=$\frac{x}{£¨x+2£©£¨x+2014£©}$£®

Answer 1

·ÖÎö £¨1£©¹éÄÉ×ܽáµÃµ½²ðÏî¹æÂÉ£¬½«Ô­Ê½±äÐκó¼ÆËã¼´¿ÉµÃµ½½á¹û£»
£¨2£©ÀûÓõóöµÄ²ðÏî¹æÂɽ«·½³Ì±äÐΣ¬Çó³ö½â£¬¼ìÑé¼´¿É£®

½â´ð ½â£º£¨1£©Ô­Ê½=1-$\frac{1}{2}$+$\frac{1}{2}$-$\frac{1}{3}$+¡­+$\frac{1}{n}$-$\frac{1}{n+1}$=1-$\frac{1}{n+1}$=$\frac{n}{n+1}$£»
£¨2£©·½³ÌÕûÀíµÃ£º$\frac{1}{x+2}$-$\frac{1}{x+3}$+$\frac{1}{x+3}$-$\frac{1}{x+4}$+¡­+$\frac{1}{x+2013}$-$\frac{1}{x+2014}$=$\frac{x}{£¨x+2£©£¨x+2014£©}$£¬
¼´$\frac{2012}{£¨x+2£©£¨x+2014£©}$=$\frac{x}{£¨x+2£©£¨x+2014£©}$£¬
È¥·ÖĸµÃ£ºx=2012£¬
¾­¼ìÑéx=2012ÊÇ·Öʽ·½³ÌµÄ½â£®
¹Ê´ð°¸Îª£º£¨1£©$\frac{n}{n+1}$

µãÆÀ ´ËÌ⿼²éÁ˽â·Öʽ·½³Ì£¬½â·Öʽ·½³ÌµÄ»ù±¾Ë¼ÏëÊÇ¡°×ª»¯Ë¼Ï롱£¬°Ñ·Öʽ·½³Ìת»¯ÎªÕûʽ·½³ÌÇó½â£®½â·Öʽ·½³ÌÒ»¶¨×¢ÒâÒªÑé¸ù£®